Average effective rents climbed 3.6 percent during the second quarter compared to a year earlier, according to REIS Inc., a real estate research data firm. Rent growth has bloomed at a fast pace since 2012, hovering around a 4 percent annual growth since then.

Strong technology markets like San Jose, Calif.; San Francisco; and Denver are seeing some of the largest increases in rental costs. In San Jose, rents rose 7.2 percent from the second quarter of 2014 to $1,951 a month. In San Francisco, rents rose 6.8 percent to $2,316. In the country's most expensive rental market, New York City, rents rose 1.7 percent over the prior quarter, reaching $3,294 a month.

Ryan Severino, senior economist at REIS, says that inadequate supplies are pushing up rental costs. "We're waiting for the supply to come on the market but it just hasn’t hit yet," Severino says.

REIS projects that 230,000 units will be completed this year -- that's nearly double above normal levels. In the second quarter, the apartment vacancy rate was just 4.2 percent.

As new supplies hit the market, "we'll probably see rent growth begin to decelerate," says Stephanie McCleskey, vice president of research for Axiometrics.